Introduction to ANSYS FLUENT

Lecture 6 Turbulence 14. 0 Release Introduction to ANSYS FLUENT 1 2011 ANSYS, Inc. January 19, 2012

Lecture Theme: Introduction The majority of engineering flows are turbulent. Successfully simulating such flows requires understanding a few basic concepts of turbulence theory and modeling. This allows one to make the best choice from the available turbulence models and near wall options for any given problem. Learning Aims: You will learn: Basic turbulent flow and turbulence modeling theory Turbulence models and near wall options available in FLUENT How to choose an appropriate turbulence model for a given problem How to specify turbulence boundary conditions at inlets Learning Objectives: You will understand the challenges inherent in turbulent flow simulation and be able to identify the most suitable model and near wall treatment for a given problem. 2 2011 ANSYS, Inc. January 19, 2012

Observation by Osborne Reynolds Flows can be classified as either : Laminar (Low Reynolds Number) Transition (Increasing Reynolds Number) Turbulent (Higher Reynolds Number) 3 2011 ANSYS, Inc. January 19, 2012

Observation by Osborne Reynolds The Reynolds number is the criterion used to determine whether the flow is laminar or turbulent The Reynolds number is based on the length scale of the flow: L x, d, d, etc. hyd Transition to turbulence varies depending on the type of flow: External flow Re L. UL. along a surface : Re X > 500 000 around on obstacle : Re L > 20 000 Internal flow : Re D > 2 300 4 2011 ANSYS, Inc. January 19, 2012

Turbulent Flow Structures A turbulent flow contains a wide range of turbulent eddy sizes Turbulent flow characteristics : Unsteady, three-dimensional, irregular, stochastic motion in which transported quantities (mass, momentum, scalar species) fluctuate in time and space Enhanced mixing of these quantities results from the fluctuations Unpredictability in detail Large scale coherent structures are different in each flow, whereas small eddies are more universal Small structures Large structures 5 2011 ANSYS, Inc. January 19, 2012

Turbulent Flow Structures Energy is transferred from larger eddies to smaller eddies (Kolmogorov Cascade) Large scale contains most of the energy In the smallest eddies, turbulent energy is converted to internal energy by viscous dissipation Energy Cascade Richardson (1922), Kolmogorov (1941) 6 2011 ANSYS, Inc. January 19, 2012

Backward Facing Step As engineers, in most cases we do not actually need to see an exact snapshot of the velocity at a particular instant. Insteadfor most problems, knowing the time averaged velocity (and intensity of the turbulent fluctuations) is all we need to know. This gives us a useful way to approach modelling turbulence. Instantaneous velocity contours Time averaged velocity contours 7 2011 ANSYS, Inc. January 19, 2012

Mean and Instantaneous Velocities If we recorded the velocity at a particular point in the real (turbulent) fluid flow, the instantaneous velocity (U) would look like this: u Fluctuating velocity Velocity U Time-average of velocity U Instantaneous velocity At any point in time: Time U U u The time average of the fluctuating velocity umust be zero: u 0 BUT, the RMS of u is not necessarily zero: u 2 0 Note you will hear reference to the turbulence energy, k. This is the sum of the 3 1 2 2 2 fluctuating velocity components: k u v w 2 8 2011 ANSYS, Inc. January 19, 2012

Overview of Computational Approaches Different approaches to make turbulence computationally tractable DNS (Direct Numerical Simulation) LES (Large Eddy Simulation) RANS (Reynolds Averaged Navier Stokes Simulation) Numerically solving the full unsteady Navier Stokes equations Resolves the whole spectrum of scales No modeling is required But the cost is too prohibitive! Not practical for industrial flows! Solves the spatially averaged N S equations Large eddies are directly resolved, but eddies smaller than the mesh are modeled Less expensive than DNS, but the amount of computational resources and efforts are still too large for most practical applications Solve time averaged Navier Stokes equations All turbulent length scales are modeled in RANS Various different models are available This is the most widely used approach for industrial flows 9 2011 ANSYS, Inc. January 19, 2012

RANS Modeling : Averaging Thus, the instantaneous Navier Stokes equations may be rewritten as Reynoldsaveraged equations: u Rij u iu i u i p u R i ij j u k t x k xi x j x (Reynolds stress tensor) j x j The Reynolds stresses are additional unknowns introduced by the averaging procedure, hence they must be modeled (related to the averaged flow quantities) in order to close the system of governing equations R uu ij i j 2 u' u' v' u' w' 2 uv ' ' v' vw ' ' 2 uw ' ' vw ' ' w' 6 unknowns 10 2011 ANSYS, Inc. January 19, 2012

RANS Modeling : The Closure Problem The Reynolds Stress tensor R must be solved ij u iuj The RANS models can be closed in two ways: Reynolds Stress Models (RSM) R ij is directly solved via transport equations (modeling is still required for many terms in the transport equations) t T u iuj uk u iuj Pij Fij Dij ij ij x k RSM is more advantageous in complex 3D turbulent flows with large streamline curvature and swirl, but the model is more complex, computationally intensive, more difficult to converge than eddy viscosity models Eddy Viscosity Models Boussinesq hypothesis Reynolds stresses are modeled using an eddy (or turbulent) viscosity, μ T R ij uu i j ui T x j u x The hypothesis is reasonable for simple turbulent shear flows: boundary layers, round jets, mixing layers, channel flows, etc. i j 2 3 T u x k k ij 2 k 3 ij Note: All turbulence models contain empiricism Equations cannot be derived from fundamental principles Some calibrating to observed solutions and intelligent guessing is contained in the models 11 2011 ANSYS, Inc. January 19, 2012

Turbulence Models Available in FLUENT RANS based models One Equation Model Spalart Allmaras Two Equation Models Standard k ε RNG k ε Realizable k ε* Standard k ω SST k ω* Reynolds Stress Model k kl ω Transition Model SST Transition Model Detached Eddy Simulation Large Eddy Simulation Increase in Computational Cost Per Iteration * Recommended choice for standard cases 12 2011 ANSYS, Inc. January 19, 2012

Two Equation Models Two transport equations are solved, giving two independent scales for calculating t Virtually all use the transport equation for the turbulent kinetic energy, k Dk Dt x j t k k x j P ; P S production dissipation Several transport variables have been proposed, based on dimensional arguments, and used for second equation. The eddy viscosity t is then formulated from the two transport variables. Kolmogorov, : t k/, l k 1/2 / k / is specific dissipation rate defined in terms of large eddy scales that define supply rate of k Chou, : t k 2 /, l k 3/2 / Rotta, l: t k 1/2 l, k 3/2 /l t 2 ( ske) S 2S ij S ij 13 2011 ANSYS, Inc. January 19, 2012

2011 ANSYS, Inc. January 19, 2012 14 Standard k- Model Equations Empirical constants determined from benchmark experiments of simple flows using air and water. ij ij t j k j S S S S x k x Dt Dk 2 ; 2 t 2 2 t 1 t C S C k x x Dt D j j k-transport equation -transport equation production dissipation 2,,, C C i k coefficients turbulent viscosity 2 k C t inverse time scale

RANS : EVM :Standard k ε (SKE) Model The Standard K Epsilon model (SKE) is the most widely used engineering turbulence model for industrial applications Model parameters are calibrated by using data from a number of benchmark experiments such as pipe flow, flat plate, etc. Robust and reasonably accurate for a wide range of applications Contains submodels for compressibility, buoyancy, combustion, etc. Known limitations of the SKE model: Performs poorly for flows with larger pressure gradient, strong separation, high swirling component and large streamline curvature. Inaccurate prediction of the spreading rate of round jets. Production of k is excessive (unphysical) in regions with large strain rate (for example, near a stagnation point), resulting in very inaccurate model predictions. 15 2011 ANSYS, Inc. January 19, 2012

RANS : EVM: Realizable k epsilon Realizable k ε (RKE) model (Shih): Dissipation rate (ε) equation is derived from the mean square vorticity fluctuation, which is fundamentally different from the SKE. Several realizability conditions are enforced for Reynolds stresses. Benefits: Accurately predicts the spreading rate of both planar and round jets Also likely to provide superior performance for flows involving rotation, boundary layers under strong adverse pressure gradients, separation, and recirculation OFTEN PREFERRED TO STANDARD K-EPSILON 16 2011 ANSYS, Inc. January 19, 2012

RANS : EVM : Spalart Allmaras (S A) Model Spalart Allmaras is a low cost RANS model solving a single transport equation for a modified eddy viscosity Designed specifically for aerospace applications involving wallbounded flows Has been shown to give good results for boundary layers subjected to adverse pressure gradients. Used mainly for aerospace and turbomachinery applications Limitations: The model was designed for wall bounded flows and flows with mild separation and recirculation. No claim is made regarding its applicability to all types of complex engineering flows. 17 2011 ANSYS, Inc. January 19, 2012

k omega Models In k models, the transport equation for the turbulent dissipation rate,, is replaced with an equation for the specific dissipation rate, The turbulent kinetic energy transport equation is still solved See Appendix for details of equation k models have gained popularity in recent years mainly because: Much better performance than k models for boundary layer flows For separation, transition, low Re effects, and impingement, k models are more accurate than k models Accurate and robust for a wide range of boundary layer flows with pressure gradient Two variations of the k model are available in FLUENT Standard k model (Wilcox, 1998) SST k model (Menter) 18 2011 ANSYS, Inc. January 19, 2012

SST Model Shear Stress Transport (SST) Model The SST model is an hybrid two equation model that combines the advantages of both k and k models k model performs much better than k models for boundary layer flows Wilcox original k model is overly sensitive to the freestream value (BC) of, while k model is not prone to such problem k- k- Wall The k e and k w models are blended such that the SST model functions like the k w close to the wall and the k e model in the freestream SST is a good compromise between k and k models 19 2011 ANSYS, Inc. January 19, 2012

RANS: Other Models in FLUENT RNG k model Model constants are derived from renormalization group (RNG) theory instead of empiricism Advantages over the standard k model are very similar to those of the RKE model Reynolds Stress model (RSM) Instead of using eddy viscosity to close the RANS equations, RSM solves transport equations for the individual Reynolds stresses 7 additional equations in 3D, compared to 2 additional equations with EVM. Much more computationally expensive than EVM and generally very difficult to converge As a result, RSM is used primarily in flows where eddy viscosity models are known to fail These are mainly flows where strong swirl is the predominant flow feature, for instance a cyclone (see Appendix) 20 2011 ANSYS, Inc. January 19, 2012

Turbulence Near the Wall The Structure of Near Wall Flows 21 2011 ANSYS, Inc. January 19, 2012

Turbulence near a Wall Near to a wall, the velocity changes rapidly. Velocity, U Distance from Wall, y If we plot the same graph again, where: Log scale axes are used The velocity is made dimensionless, from U/U ( ) The wall distance vector is made dimensionless Then we arrive at the graph on the next page. The shape of this is generally the same for all flows: 22 2011 ANSYS, Inc. January 19, 2012

Turbulence Near the Wall By scaling the variables near the wall the velocity profile data takes on a predictable form (transitioning from linear to logarithmic behavior) Scaling the non-dimensional velocity and non-dimensional distance from the wall results in a predictable boundary layer profile for a wide range of flows Linear Logarithmic Since near wall conditions are often predictable, functions can be used to determine the near wall profiles rather than using a fine mesh to actually resolve the profile These functions are called wall functions 23 2011 ANSYS, Inc. January 19, 2012

Choice of Wall Modeling Strategy. In the near wall region, the solution gradients are very high, but accurate calculations in the near wall region are paramount to the success of the simulation. The choice is between: Resolving the Viscous Sublayer First grid cell needs to be at about y + = 1 This will add significantly to the mesh count Use a low Reynolds number turbulence model (like k omega) Generally speaking, if the forces on the wall are key to your simulation (aerodynamic drag, turbomachinery blade performance) this is the approach you will take Using a Wall Function First grid cell needs to be 30 < y + < 300 (Too low, and model is invalid. Too high and the wall is not properly resolved.) Use a wall function, and a high Re turbulence model (SKE, RKE, RNG) Generally speaking, this is the approach if you are more interested in the mixing in the middle of the domain, rather than the forces on the wall 24 2011 ANSYS, Inc. January 19, 2012

Turbulence Near the Wall Fewer nodes are needed normal to the wall when logarithmic based wall functions are used (compared to more detailed low Re wall modeling) y u y u Logarithmic-based Wall functions used to resolve boundary layer Near-wall resolving approach used to resolve boundary layer Boundary layer First node wall distance is reflected by y+ value 25 2011 ANSYS, Inc. January 19, 2012

Example in Predicting Near wall Cell Size During the pre processing stage, you will need to know a suitable size for the first layer of grid cells (inflation layer) so that Y + is in the desired range. The actual flow field will not be known until you have computed the solution (and indeed it is sometimes unavoidable to have to go back and remesh your model on account of the computed Y + values). To reduce the risk of needing to remesh, you may want to try and predict the cell size by performing a hand calculation at the start. For example: Air at 20 m/s = 1.225 kg/m 3 = 1.8x10-5 kg/ms Flat plate, 1m long For a flat plate, Reynolds number ( Re VL l ) gives Re l = 1.4x10 6 (Recall from earlier slide, flow over a surface is turbulent when Re L > 5x10 5 ) y The question is what height (y) should the first row of grid cells be. We will use SWF, and are aiming for Y + 50 26 2011 ANSYS, Inc. January 19, 2012

Example in Predicting Near wall Cell Size [2] Begin with the definition of y + and rearrange: U y y y y U The target y + value and fluid properties are known, so we need U, which is defined as: U w The wall shear stress, w,can be found from the skin friction coefficient, C f : w C 2 f U A literature search suggests a formula for the skin friction on a plate 1 thus: C f 1 2 0.058Re 0.2 l Re is known, so use the definitions to calculate the first cell height 0.2 C f 0.058 Rel.0034 1 w 2 2 U 0.83 kg/ m s We know we are aiming for y + of 50, hence: y -4 y 9x10 m U our first cell height y should be approximately 1 mm. 27 2011 ANSYS, Inc. January 19, 2012 U 2 C f w 0.82 m/s 1 An equivalent formula for internal flows, with Reynolds number based on the pipe diameter is C f = 0.079 Re d 0.25

Limitations of Wall Functions In some situations, such as boundary layer separation, logarithmic based wall functions do not correctly predict the boundary layer profile Wall functions applicable Wall functions not applicable Non-equilibrium wall functions have been developed in FLUENT to address this situation but they are very empirical. A more rigorous approach is recommended if affordable In these cases logarithmic based wall functions should not be used Instead, directly resolving the boundary layer can provide accurate results 28 2011 ANSYS, Inc. January 19, 2012

Enhanced Wall Treatment (EWT) Need for y+ insensitive wall treatment EWT smoothly varies from low Re to wall function with mesh resolution EWT available for k and RSM models Similar approach implemented for k equation based models 29 2011 ANSYS, Inc. January 19, 2012

Choosing a near Wall Treatment Standard Wall Functions The Standard Wall Function options is designed for high Re attached flows The near wall region is not resolved Near wall mesh is relatively coarse Non Equilibrium Wall Functions For better prediction of adverse pressure gradient flows and separation Near wall mesh is relatively coarse Enhanced Wall Treatment * Used for low Re flows or flows with complex near wall phenomena Generally requires a very fine near wall mesh capable of resolving the near wall region Can also handle coarse near wall mesh User Defined Wall Functions Can host user specific solutions * Recommended choice for standard cases 30 2011 ANSYS, Inc. January 19, 2012

y+ for the SST and k omega Models The SST and k models were formulated to be near wall resolving models where the viscous sublayer is resolved by the mesh To take full advantage of this formulation, y+ should be < 2 This is necessary for accurate prediction of flow separation These models can still be used with a coarser near wall mesh and produce valid results, within the limitations of logarithmic wall functions The first grid point should still be in the logarithmic layer (y+ < 300) Many advantages of these models may be lost when a coarse near wall mesh is used 31 2011 ANSYS, Inc. January 19, 2012

Inlet Boundary Conditions When turbulent flow enters a domain at inlets or outlets (backflow), boundary ' ' conditions for k, ε, ω and/or u i u j must be specified, depending on which turbulence model has been selected Four methods for directly or indirectly specifying turbulence parameters: 1) Explicitly input k, ε, ω, or Reynolds stress components (this is the only method that allows for profile definition) Note by default, the FLUENT GUI enters k=1 m²/s² and =1m²/s³. These values MUST be changed, they are unlikely to be correct for your simulation. 2) Turbulence intensity and length scale Length scale is related to size of large eddies that contain most of energy For boundary layer flows: l 0.4δ 99 For flows downstream of grid: l opening size 3) Turbulence intensity and hydraulic diameter (primarily for internal flows) 4) Turbulence intensity and viscosity ratio (primarily for external flows) 32 2011 ANSYS, Inc. January 19, 2012

Inlet Turbulence Conditions If you have absolutely no idea of the turbulence levels in your simulation, you could use following values of turbulence intensities and viscosity ratios: Usual turbulence intensities range from 1% to 5% The default turbulence intensity value of 0.037 (that is, 3.7%) is sufficient for nominal turbulence through a circular inlet, and is a good estimate in the absence of experimental data For external flows, turbulent viscosity ratio of 1 10 is typically a good value For internal flows, turbulent viscosity ratio of 10 100 it typically a good value For fully developed pipe flow at Re = 50,000, the turbulent viscosity ratio is around 100 33 2011 ANSYS, Inc. January 19, 2012

RANS Turbulence Model Usage Model Spalart-Allmaras Standard k ε Realizable k ε* RNG k ε Standard k ω SST k ω* RSM Behavior and Usage Economical for large meshes. Performs poorly for 3D flows, free shear flows, flows with strong separation. Suitable for mildly complex (quasi-2d) external/internal flows and boundary layer flows under pressure gradient (e.g. airfoils, wings, airplane fuselages, missiles, ship hulls). Robust. Widely used despite the known limitations of the model. Performs poorly for complex flows involving severe pressure gradient, separation, strong streamline curvature. Suitable for initial iterations, initial screening of alternative designs, and parametric studies. Suitable for complex shear flows involving rapid strain, moderate swirl, vortices, and locally transitional flows (e.g. boundary layer separation, massive separation, and vortex shedding behind bluff bodies, stall in wide-angle diffusers, room ventilation). Offers largely the same benefits and has similar applications as Realizable. Possibly harder to converge than Realizable. Superior performance for wall-bounded boundary layer, free shear, and low Reynolds number flows. Suitable for complex boundary layer flows under adverse pressure gradient and separation (external aerodynamics and turbomachinery). Can be used for transitional flows (though tends to predict early transition). Separation is typically predicted to be excessive and early. Offers similar benefits as standard k ω. Dependency on wall distance makes this less suitable for free shear flows. Physically the most sound RANS model. Avoids isotropic eddy viscosity assumption. More CPU time and memory required. Tougher to converge due to close coupling of equations. Suitable for complex 3D flows with strong streamline curvature, strong swirl/rotation (e.g. curved duct, rotating flow passages, swirl combustors with very large inlet swirl, cyclones). * Recommended choice for standard cases 34 2011 ANSYS, Inc. January 19, 2012

RANS Turbulence Model Descriptions Model Spalart Allmaras Standard k ε RNG k ε Realizable k ε Standard k ω SST k ω RSM Description A single transport equation model solving directly for a modified turbulent viscosity. Designed specifically for aerospace applications involving wall-bounded flows on a fine near-wall mesh. FLUENT s implementation allows the use of coarser meshes. Option to include strain rate in k production term improves predictions of vortical flows. The baseline two-transport-equation model solving for k and ε. This is the default k ε model. Coefficients are empirically derived; valid for fully turbulent flows only. Options to account for viscous heating, buoyancy, and compressibility are shared with other k ε models. A variant of the standard k ε model. Equations and coefficients are analytically derived. Significant changes in the ε equation improves the ability to model highly strained flows. Additional options aid in predicting swirling and low Reynolds number flows. A variant of the standard k ε model. Its realizability stems from changes that allow certain mathematical constraints to be obeyed which ultimately improves the performance of this model. A two-transport-equation model solving for k and ω, the specific dissipation rate (ε / k) based on Wilcox (1998). This is the default k ω model. Demonstrates superior performance for wall-bounded and low Reynolds number flows. Shows potential for predicting transition. Options account for transitional, free shear, and compressible flows. A variant of the standard k ω model. Combines the original Wilcox model for use near walls and the standard k ε model away from walls using a blending function. Also limits turbulent viscosity to guarantee that τ T ~ k. The transition and shearing options are borrowed from standard k ω. No option to include compressibility. Reynolds stresses are solved directly using transport equations, avoiding isotropic viscosity assumption of other models. Use for highly swirling flows. Quadratic pressure-strain option improves performance for many basic shear flows. 35 2011 ANSYS, Inc. January 19, 2012

Summary Turbulence Modeling Guidelines Successful turbulence modeling requires engineering judgment of: Flow physics Computer resources available Project requirements Accuracy Turnaround time Choice of Near wall treatment Modeling procedure 1. Calculate characteristic Reynolds number and determine whether flow is turbulent. 2. If the flow is in the transition (from laminar to turbulent) range, consider the use of one of the turbulence transition models (not covered in this training). 3. Estimate wall adjacent cell centroid y + before generating the mesh. 4. Prepare your mesh to use wall functions except for low Re flows and/or flows with complex near wall physics (non equilibrium boundary layers). 5. Begin with RKE (realizable k ε) and change to S A, RNG, SKW, or SST if needed. Check the tables on previous slides as a guide for your choice. 6. Use RSM for highly swirling, 3 D, rotating flows. 7. Remember that there is no single, superior turbulence model for all flows! 36 2011 ANSYS, Inc. January 19, 2012

Appendix 37 2011 ANSYS, Inc. January 19, 2012

Example #1 Turbulent Flow Past a Blunt Flat Plate Turbulent flow past a blunt flat plate was simulated using four different turbulence models. 8,700 cell quad mesh, graded near leading edge and reattachment location. Non equilibrium boundary layer treatment U 0 x R Re 50,000 D D Recirculation zone Reattachment point N. Djilali and I. S. Gartshore (1991), Turbulent Flow Around a Bluff Rectangular Plate, Part I: Experimental Investigation, JFE, Vol. 113, pp. 51 59. 38 2011 ANSYS, Inc. January 19, 2012

Example #1 Turbulent Flow Past a Blunt Flat Plate Contours of Turbulent Kinetic Energy (m 2 /s 2 ) 0.70 0.63 0.56 0.49 0.42 0.35 0.28 0.21 0.14 0.07 0.00 Standard k ε Realizable k ε RNG k ε Reynolds Stress 39 2011 ANSYS, Inc. January 19, 2012

Example #1 Turbulent Flow Past a Blunt Flat Plate Predicted separation bubble: Standard k ε (SKE) Skin Friction Coefficient C f 1000 Realizable k ε (RKE) SKE severely underpredicts the size of the separation bubble, while RKE predicts the size exactly. Experimentally observed reattachment point is at x / D = 4.7 Distance Along Plate, x / D 40 2011 ANSYS, Inc. January 19, 2012

Example #2 : Pipe Expansion with Heat Transfer Reynolds Number Re D = 40750 Fully Developed Turbulent Flow at Inlet Experiments by Baughn et al. (1984) H. q=0 D q=const Outlet Inlet d axis H 40 x H 41 2011 ANSYS, Inc. January 19, 2012

Example #2 : Pipe Expansion with Heat Transfer Plot shows dimensionless distance versus Nusselt Number Best agreement is with SST and k omega models which do a better job of capturing flow recirculation zones accurately 42 2011 ANSYS, Inc. January 19, 2012

Example #3 Turbulent Flow in a Cyclone 40,000 cell hexahedral mesh 0.1 m 0.12 m High order upwind scheme was used. U in = 20 m/s Computed using SKE, RNG, RKE and RSM (second moment closure) models with the standard wall functions 0.97 m 0.2 m Represents highly swirling flows (W max = 1.8 U in ) 43 2011 ANSYS, Inc. January 19, 2012

Example #3 Turbulent Flow in a Cyclone Tangential velocity profile predictions at 0.41 m below the vortex finder 44 2011 ANSYS, Inc. January 19, 2012

Example 4: Diffuser Shear Stress Transport (SST) Model It accounts more accurately for the transport of the turbulent shear stress, which improves predictions of the onset and the amount of flow separation compared to k models Standard k- fails to predict separation SST result and experiment Experiment Gersten et al. 45 2011 ANSYS, Inc. January 19, 2012

Turbulent Flow Structures Related to k and Characteristics of the Turbulent Structures: Length scale : l [m] Velocity scale : k [m/s] Time scale : l [s] k Shape (non isotropic larger structures) 1 k u' v' w' 2 2 2 2 - Turbulent kinetic energy : [m 2 /s 2 ] - Turbulent kinetic energy dissipation : [m 2 /s 3 ] ~ k 3/2 /l(dimensional analysis) - Turbulent Reynolds : Re t = k 1/2.l/ ~ k 2 / [-] u 1 I U U 2k 3 - Turbulent Intensity : [-] u x, t U x, t u x t i i i, Instantaneous component Time average component Fluctuating component 46 2011 ANSYS, Inc. January 19, 2012

2011 ANSYS, Inc. January 19, 2012 47 k models are RANS two equations based models One of the advantages of the k formulation is the near wall treatment for low Reynolds number computations designed to predict correct behavior when integrated to the wall the k models switches between a low Reynolds number formulation (i.e. direct resolution of the boundary layer) at low y + values and a wall function approach at higher y + values while Low Reynolds number variations of standard k models use damping functions to attempt to reproduce correct near wall behavior k omega Model j t j j i ij j k t j j i ij t x x f x u k Dt D x k x k f x u Dt Dk k 2 ω = specific dissipation rate 1 k ij ij k k i j j i j i ij k x u x u x u u u R 3 2 3 2 T T